MA5209 Algebraic Topology Wayne Lawton Department of Mathematics National University of Singapore S , Lecture 6. Applications Degrees of Maps of Spheres, Antipodal Preserving Maps Borsuk-Ulam and Lusternik-Schnirelmann Theorems Ham Sandwich Theorem Euler-Poincare and Hopf Trace Formulae Lefschetz Fixed Point and Lefschetz-Hopf Index Formula Tor Functor and the Universal Coefficient Theorem Hopf’s Homotopy Theorem (27, 30 October 2009)

Maps of Spheres Definition The degree of a map such that is Questions Let Prove that Isa homotopy invariant? For eachand construct such that Ifand then areandnecessarily homotopic? Show that Explain the geometric meaning of is a simplicial map.if so

Triangulation of Spheres Definition Simplicial complexis the boundary of the convex set spanned by the set where is the standard basis for Figure n = 1 Questions Show that The set of q-simplices inis How many q-simplices are there? triangulation

Triangulation of Spheres Lemma The chaindefined by generates Proof where Question Show that every chain is a multiple of

Antipodal Maps Theorem simplicial map defined by its values on vertices Definition The antipodal map Proof Then is the satisfies induces the chain map and

Antipodal Maps Corollary A map homotopic to the antipodal map Lemma A map Proof Construct the homotopy with no fixed points is degree with no fixed points has Corollary Ifis even andis homotopic tothenhas a fixed point.

Vector Fields on Spheres vector field Theorem There exists a nonvanishing tangent is odd.if and only if Proof Letbe a nonvanishing tangent is a homotopy fromtoSince defined by vector field and let Since has no fixed points (why) For oddthe converse is proved by constructing

Antipodal Preserving Maps Define by and by its restriction, also denoted by Definition A map and the antipodal map on or is antipodal preserving if Lemma If Example is antipodal preserving then there existsand a simplicial approximation which is antipodal preserving. to

Antipodal Preserving Maps Proof It follows from Lebesgue’s Lemma that such that andSince Partition there exists so and for everydefineso that and and define

Homology with Coefficients in a Group Definition Ifis a simplicial complex and chain complex where for every denote the is the group of is an abelian group we let simplices in tensor products as since both of valued functions on the set This can be defined using andaremodules. Heuristically, we can obtainfrom by replacing integer coefficients of simplices by elements inand letting the boundary operator belinear.Homology groups are defined as usual and denoted by

Antipodal Preserving Maps Theorem If is odd.is an antipodal map then or Proof It suffices to prove that Let is an isomorphism. sum of all k-simplices in Question: prove SinceAssume that andwhere Then whereis the sum of simplices incontaining vertex

Antipodal Preserving Maps Then apply of simplices each which either contains ).and does not contain where or (contains and this is impossible -why? Repeat to obtain to obtain is a sum

Antipodal Preserving Maps Lemma Ifis antipodal preserving Proof Else then Letis contractible However since Since and it follows that is antipodal preserving, is odd. This contradiction concludes the proof.

Borsuk-Ulam Theorem Theorem (BU) For every map such thatthere exists Proof Else the map is antipodal - contradicting the previous Lemma. defined by

Lusternik-Schnirelmann Theorem Theorem (LS) If with closed then contains a pair of antipodal points.one of the Proof Since the functiondefined by is continuous there exists Hence iffor somethencontains the pair of antipodal pointsOtherwise and this concludes the proof.

Ham Sandwich Theorem Theorem (HS) If are bounded measurable subsets ofthere exists a hyperplane that bisects each Proof Define by where is continuous so the Borsuk-Ulam Theorem implies that there exists Clearlyso is a hyperplane that bisects each

Euler-Poincare Formula Definition Euler characteristic of a chain complex that satisfies Theorem is and Betti numbers are Proof Smith normal form for integer matrices 

Rational Homology is a simplicial complex withLemma If then and Proof Follows sinceis a field so that all the modules are vector spaces over

Hopf Trace Theorem Theorem If is a chain map then Remarkgives the Euler-Poincare Formula Proof Choose bases Theorem (HT) If is a simplicial complex and for eachand for each basis element letdenote the coefficient ofin the expansion in this basis forTherefore Since so the formula follows by summing.

Lefschetz Fixed-Point Theorem Theorem If map Theorem (LFP) If Definition The Lefschetz number of a triangulable top. space is of a thenhas a FP. Proof Else simp. coml. and simp. approx. toand subdivision chain map gives For oriented q-simplices The chain map inand‘in’ soand do not lie in same simp. of henceso has coef. zero in

Lefschetz-Hopf Index Formula manifold with isolated fixed points Theorem (LH) If is a map of a closed where the index then is defined of an isolated fixed point by where is defined by where we identify points other than with a closed neighborhood ofthat has no fixed Proof Henle p249 proves simple case of result in H. Hopf, U¨ber die algebraische Anzahl von Fixpunkten,Math. Z. 29 (1929), 493–524. Also see R. F. Brown, The Lefschetz Fixed Point Theorem, 1971 and Fixed point theory, History of Topology, 1999, pp. 271–299.

Universal Coefficient Theorem for Homology Theorem (UCTH) Letbe a fee chain complex (considered as amodule) and be an abelian group. Then there exists a exact sequence that splits, so Proof p in Spanier’s Algebraic Topology Remark Tor is a covariant functor in each argument Proof p in Munkres’ Elem. of Alg. Topology Example Ifis simplicial chain complex of a Klein bottle thenhence

Categorical Considerations Tor functors. The category of left R-modules also has enough projectives. If A is a fixed right R-module, then the tensor product with A gives a right exacttensor product covariant functor on the category of left R-modules; its left derivatives are the Tor functorsTor functors TorRi(A,B). Ext functors. If R is a ring, then the category of all left R-modules is an abelianringR-modules category with enough injectives. If A is a fixed left R-module, then the functor Hom(A,-) is left exact, and its right derived functors are the Ext functorsExt functors ExtRi(A,B). The Tor functor as well as the Ext functor that arises in cohomology, are examples of derived functors in abelian categories, here are several websites to learn more:

Hopf’s Homotopy Theorem Theorem (HH) Two maps of a sphere to itself are homotopic if and only if they have the same degree. Proof See p. 340, in Dugundji’s Topology.